## Fillers

### Quick Start

When you add a filler to a PSA (or any other polymer or gel network), the G' goes up - which may be good or bad. For low levels, the effect on G' can be estimated via Guth-Smallwood. At higher levels other effects take over.

Start by choosing the G'_{0} of the unfilled material and a volume fraction φ of the filler, and look at G'_{f} of the filled material. The other input and outputs are discussed below.

### Fillers

_{0}MPa

_{m}

_{f}MPa

_{f}g MPa

_{f}d MPa

_{f}dx MPa

Many polymers and gel networks contain fillers. Sometimes these form some positive function, othertimes they are a cheap padding. In either case they affect the modulus, G', of the material. This app started as being specific to PSA because G' is especially important within such formulations (Dahlquist). Whatever your application, it is useful to know how the filler affects the modulus.

In the absence of any specific knowledge about special filler/formulation interactions, then for low filler levels, the classic formula for the effect of particles is Guth-Smallwood (or Guth-Gold). The modulus G'_{f} when filled with a volume fraction φ_{f} compared to the original modulus G'_{0} is given by:

G'_{f}/G'_{0} = 1 + 2.5φ_{f} + 14.1φ_{f}^{2}

If the particle has an aspect ratio (length/width) of g then the formula changes to:

G'_{f}/G'_{0} = 1 + 0.67g.φ_{f} + 1.62(g.φ_{f})^{2}

Use the calculations as a rough guide to your own system, and refine it with any extra information you can find. The values with and without the aspect ratio are shown in the two outputs. As they are both approximations, they don't precisely overlap when g=1.

There is evidence^{1} that the model understimates the effects at higher loadings and a more satisfactory equation (for spherical particles) is:

G'_{f}/G'_{0} =(1-φ_{f}/φ_{m})^{-n}

φ_{m} is the fraction where "interesting" things start to happen and is typiclaly ~0.58 and n is typically 1.8. The value shown as G'_{f} d, where d stands for "diverges".

So far, the filler has been "neutral" in that it doesn't interact strongly with the polymer network. If you get an extra fraction x of links via strong particle-polymer interactions, then

G'_{f}/G'_{0} =(1+x)(1-φ_{f}/φ_{m})^{-n}

This value is shown as G'_{f} dx, where df stands for "diverges with extra links"

Note: These equations imply, as is found in practice, the surprising fact that particle size has no influence on the modulus effect. The formulator's instinct will be that smaller particles are generally better in terms of strength (less crack defect sites) and looks (less ugly/hazy).

^{1}

P.L. Drzal and K.R. Shull,
*Elasticity, fracture and thermoreversible gelation of highly filled physical gels*, Eur. Phys. J. E 17, 477-483 (2005)